Geometric Conformal Prediction with Spatial Ranks and Multivariate Quantiles

Anton Conrad, Eric Moulines, Julien Perez
Proceedings of the 43rd International Conference on Machine Learning, PMLR 306:21323-21348, 2026.

Abstract

In multi-target regression and multi-class classification, uncertainty is inherently multivariate: prediction regions must capture joint dependencies across correlated outputs. Conformal prediction provides distribution-free guarantees, yet extending it to vector-valued outputs remains challenging—scalar aggregation discards geometric structure, while optimal transport (OT) approaches are computationally demanding and sensitive to outliers. We introduce two conformal methods based on geometric quantiles and spatial ranks: Geometric Conformalized Quantile Regression (GCQR) constructs prediction regions from learned conditional geometric quantiles, while Geometric Rank Conformal Prediction (GRCP) uses the radial rank of vector-valued conformity scores as the nonconformity measure. We propose multiple estimators offering different tradeoffs between computational cost and adaptivity to feature-dependent heterogeneity, with scalable learning via partially input-convex neural networks. On multi-target regression and multi-class classification benchmarks, GCQR and GRCP attain near-nominal coverage with consistently tighter prediction regions than scalarized and multivariate baselines.

Cite this Paper

BibTeX
@InProceedings{pmlr-v306-conrad26a, title = {Geometric Conformal Prediction with Spatial Ranks and Multivariate Quantiles}, author = {Conrad, Anton and Moulines, Eric and Perez, Julien}, booktitle = {Proceedings of the 43rd International Conference on Machine Learning}, pages = {21323--21348}, year = {2026}, editor = {Zhang, Tong and Dudik, Miroslav and Jaggi, Martin and Agarwal, Alekh and Li, Sharon and Schuurmans, Dale and Zhu, Jerry and Berkenkamp, Felix and Dong, Hanze and Bietti, Alberto}, volume = {306}, series = {Proceedings of Machine Learning Research}, month = {06--11 Jul}, publisher = {PMLR}, pdf = {https://raw.githubusercontent.com/mlresearch/v306/main/assets/conrad26a/conrad26a.pdf}, url = {https://proceedings.mlr.press/v306/conrad26a.html}, abstract = {In multi-target regression and multi-class classification, uncertainty is inherently multivariate: prediction regions must capture joint dependencies across correlated outputs. Conformal prediction provides distribution-free guarantees, yet extending it to vector-valued outputs remains challenging—scalar aggregation discards geometric structure, while optimal transport (OT) approaches are computationally demanding and sensitive to outliers. We introduce two conformal methods based on geometric quantiles and spatial ranks: Geometric Conformalized Quantile Regression (GCQR) constructs prediction regions from learned conditional geometric quantiles, while Geometric Rank Conformal Prediction (GRCP) uses the radial rank of vector-valued conformity scores as the nonconformity measure. We propose multiple estimators offering different tradeoffs between computational cost and adaptivity to feature-dependent heterogeneity, with scalable learning via partially input-convex neural networks. On multi-target regression and multi-class classification benchmarks, GCQR and GRCP attain near-nominal coverage with consistently tighter prediction regions than scalarized and multivariate baselines.} }
Endnote
%0 Conference Paper %T Geometric Conformal Prediction with Spatial Ranks and Multivariate Quantiles %A Anton Conrad %A Eric Moulines %A Julien Perez %B Proceedings of the 43rd International Conference on Machine Learning %C Proceedings of Machine Learning Research %D 2026 %E Tong Zhang %E Miroslav Dudik %E Martin Jaggi %E Alekh Agarwal %E Sharon Li %E Dale Schuurmans %E Jerry Zhu %E Felix Berkenkamp %E Hanze Dong %E Alberto Bietti %F pmlr-v306-conrad26a %I PMLR %P 21323--21348 %U https://proceedings.mlr.press/v306/conrad26a.html %V 306 %X In multi-target regression and multi-class classification, uncertainty is inherently multivariate: prediction regions must capture joint dependencies across correlated outputs. Conformal prediction provides distribution-free guarantees, yet extending it to vector-valued outputs remains challenging—scalar aggregation discards geometric structure, while optimal transport (OT) approaches are computationally demanding and sensitive to outliers. We introduce two conformal methods based on geometric quantiles and spatial ranks: Geometric Conformalized Quantile Regression (GCQR) constructs prediction regions from learned conditional geometric quantiles, while Geometric Rank Conformal Prediction (GRCP) uses the radial rank of vector-valued conformity scores as the nonconformity measure. We propose multiple estimators offering different tradeoffs between computational cost and adaptivity to feature-dependent heterogeneity, with scalable learning via partially input-convex neural networks. On multi-target regression and multi-class classification benchmarks, GCQR and GRCP attain near-nominal coverage with consistently tighter prediction regions than scalarized and multivariate baselines.
APA
Conrad, A., Moulines, E. & Perez, J.. (2026). Geometric Conformal Prediction with Spatial Ranks and Multivariate Quantiles. Proceedings of the 43rd International Conference on Machine Learning, in Proceedings of Machine Learning Research 306:21323-21348 Available from https://proceedings.mlr.press/v306/conrad26a.html.

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